Can I screw cement board directly over my old plaster and lath kitchen wall to prepare for tile?

Don't fasten cement board over plaster and lath because the old wall will flex and crack your new tile. Strip the section down to the bare studs using a pry bar, then drive cement board screws directly into the framing. If you expose old knob-and-tube or nick wiring during demo, stop and hire a licensed electrician to update the circuit before closing the wall.